Bose-Einstein Condensation and

Superfluidity

Gordon BaymUniversity of Illinois, Urbana

January 2004

東京大学

Fermions (Fermi-Dirac, 1926): Particles that obey the exclusion principle(Pauli, 1925). Can’t put two in same state at the same time.

Bosons (Bose-Einstein, 1924-5): Particles that don’t obey the exclusion principle. Can put many in the same state at the same time

A. Einstein

S. Bose

S.N. Bose 1924: concept of light quanta as particles with2 polarization states. New statistics => Planck distribution:

A. Einstein 1924: Extension to monoatomic ideal gases:

Condensation:

Condensate

I maintain that in this case a number of molecules steadily growing with increasing density goes over in the first quantum state ... a separation is affected; one part ‘condenses,’ the rest remains a saturated ideal gas.

A. Einstein, 1925

Bose-Einstein Condensation

Hot atoms (bosons) in a box

Cool below Bose-Einsteintransition temperature

At absolute zero temperature motion “ceases”

Bose-Einstein condensateGravity

Free Bose gas

Box Potential well (trap)

In condensed system have macroscopic occupation of single (generally lowest) mode

: ground state

: flow state (vortex)

MANY-PARTICLE WAVE FUNCTION

= condensate wave function

FINITE TEMPERATURE

No. condensed particles

Thermal wavelength

Which “statistics” apply to nature? i.e., is ordinary matter made of

fermions or bosons?

The [Fermi-Dirac] solution ... is probably the correct one for gas molecules, since it is known to be the correct one for electrons in an atom, and one would expect molecules to resemble electrons more closely than light quanta.

P.A.M. Dirac, 1926

With a heavy heart I have become converted to the idea that Fermi … , not Einstein-Bose, is the correct statistics [for electrons].

W. Pauli to E. Schrödinger, Nov. 1926

Superfluid 4He: The first Bose-Einstein condensate

W.H. Keesom and Miss A.P. Keesom (1935): specific heat of liquid helium

F.London (1938): Spectroscopic data => 4He obeys Bose-Einstein statistics:

“The strange change of state of liquid helium at 2.19 o abs., even though it occurs in the liquid and not in the gaseous state, is due to the condensation mechanism of the Bose-Einstein gas.”

“It seems difficult not to imagine a connexion withthe condensation phenomenon of the Bose-Einstein statistics.” (London, 1938)

Superfluid Liquid Helium

Flows through tiny capillaries without friction

Flows around a closed pipe forever

Temperatures below “Lambda point” 2.17o above absolute zero

1938

(Tony Leggett)

Spin bucket of superfluid

helium slowly. Heliumliquid remains at rest!

Spin fast enough.Form vortex in center

of liquid!

L. Landau (1941): rejects suggestion “thathelium-II should be considered as a degenerate ideal Bose gas.” Importance of interactions!

ROLE OF STATISTICS:

Sydoriak, Grilley, and Hammel (1948) liquified 3He.

Osborne, Winstock, and Abraham (1948): no superflow down to 1.05 K.

Bose character critical to superfluidity

Order parameter of Bose-condensed system

-- 0 in normal system-- constant in BEC

= complex order parameter

Free particle state, |N>

If |N> and |N-1> differ only by number of particles incondensate then

In weakly interacting Bose gas:

Time dependent order parameter

condensate wave function

condensate density

superfluid velocity

chemical potential

superfluid acceleration eqn.

Equilibrium:

Flow and superfluidity

Complex order parameter: => flow

Superfluid velocity

Superfluid mass density =Normal mass density =

At T=0 in 4He, s = , n0 = 0.09 nCondensate density differs from superfluid mass density:

Momentum density of superfluid flow = s vs

BOSE CONDENSED SYSTEMS

Low temperature systems of bosons: liquid 4He trapped bosonic atoms excitons in semiconductors (?)

Nuclear matter pion condensation kaon condensation

Vacuum as Bose condensed state Chiral symmetry breaking Gluon condensation Higgs condensation Graviton condensation, g

PION CONDENSED MATTER

Softening of collective spin-isospin oscillation of nuclear matter

Above critical density have transition to new state withnucleons rotated in isospin space:

with formation of macroscopic pion field

Important, if it exists, for enhanced cooling of neutron stars by neutrino emission

Transition density very sensitive to effective particle-hole interactions (Landau g’) and -hole interactions

Analogous neutral pion condensate

can coexist with

STRANGENESS (KAON) CONDENSATES

Analogous to condensateChiral SU(3) X SU(3) symmetry of strong interactions => effective low energy interaction

Kaplan and Nelson (1986),Brown et al. (1994)

“Effective mass” term lowers K energies in matter

=> condensation

Rotate u and s quark states:

Form condensate

Admix in n; in p

Results very sensitive to K-

interactions in matter (Pandharipande, Pethick and Vesteinn, 1995) -

* Would soften equation of state and lower maximumneutron star mass to ~ 1.5 solar masses* Would enhance neutrino luminosity and cooling of neutronstars

Can also form condensate => macroscopic η field

Condensates in vacuum

EXPERIMENTAL BOSE-EINSTEIN DECONDENSATION

Ultrarelativistic heavy ion collisions:2000: RHIC 100 GeV/A + 100 GeV/A colliding beams2007?: LHC 2600 GeV/A + 2600 GeV/A

Relativistic Heavy Ion Collider (RHIC) (Brookhaven, NY)

Break chiral symmetry indifferent state? (Disorderedchiral condensate?)

N~104, V ~ 103 fm3 : - BEC unlikely; entropy too high

Applications in Biology

R. Penrose, Shadows of the Mind (1994)

A strong proponent of the idea thatBose-Einstein condensation may provide the “unitary sense of self” that seems to be characteristic of consciousness, in relation to Fröhlich’s ideas is Ian Marshall (1989) …

Application to the MoviesInformation, Adaptive Contracting, Distributional Dynamics, Bayesian Choice, Bose-Einstein Statisticsand the Movies